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In mathematics it is generally not allowed to change order of limits. For example it is not always true for a sequence of functions $f_n$, that $\int_a^b \left(\sum_{n=0}^\infty f_n(x) \right) dx = \sum_{n=0}^\infty \left(\int_a^b f_n(x) dx\right)$. (Note that series $\sum_{n=0}^\infty\ldots$ and the integral $\int_a^b \ldots dx$ are mathematically defined via limits of sequences).

In my experience it happens a lot in physics lectures, that limits are changed in their order without any additional comment (such as mentioning Fubini's theorem or uniform convergence). It also seems to me that there are not many examples relevant for physics where changing the order of limits yield wrong results.

I'm looking for good physical examples showing to students that one has to be careful when he changes the order of limits. So for which physical example the order of the limits is important and you get a wrong result, when you change it?

bobie
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Stephan Kulla
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  • You're not changing the limits. You're changing the order of operators which is only valid for linear operators (or commuting operators). – tpg2114 Dec 14 '14 at 21:58
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    In physics lectures, you do *not* change the limits so as to get the wrong results because you don't actually want to present the wrong result to the class. – Peter Shor Dec 14 '14 at 22:00
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    @tpg2114 The post specifically mentions that both operators are defined via limits. Anyway, you should find plenty of examples in statistical mechanics: In particular you'll see a lot of limits where the ratio has to be constant (both the size of a system and the number of particles it contains has to go to infinity, yet their ratio, density, has to remain constant) – alarge Dec 14 '14 at 22:02
  • @alarge But the limits aren't changing -- OP isn't saying that instead of the integral from $a$ to $b$ it's now $a$ to $c \neq b$. That would be changing the limits. Instead, the question is about changing the order of the operators (or order of the limits) which is not the same as changing the limits. – tpg2114 Dec 14 '14 at 22:03
  • @tpg2114: By "changing the limit" I mean "changing the order of the limits" ;-) I will correct this mistake... – Stephan Kulla Dec 14 '14 at 22:10
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    This question (v2) seems like a list question. – Qmechanic Dec 15 '14 at 00:03
  • This appears to be too broad of a question because there is not really any wrong answer. – Kyle Kanos Dec 15 '14 at 03:58
  • One such situation came up in chat recently. DanielSank described an LC circuit ladder, where any finite segment has an imaginary impedance, but the whole thing seems to have a real impedance. The resolution is that there is another limit involved in calculating these quantities, and the order of limits (response time, number of segments) is tacitly switched between the two cases. –  Feb 24 '15 at 04:31

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The low-frequency ($\omega\rightarrow 0$), long-wavelength ($q\rightarrow 0$) conductivity of an electron gas in the random phase approximation depends on the order in which those two limits are taken.

Intuitively, if you take the $\omega\rightarrow 0$ limit first, you're talking about a static, long-wavelength potential to which the electrons adjust, so the conductivity is imaginary (non-dissapative). If you take the $q\rightarrow 0$ limit first, you're talking about a uniform, slowly varying field applied to the gas, so you get a current (real conductivity). See page 27 of these notes for a discussion.

Basically, you order the limits one way to learn about Thomas-Fermi screening, and the other way to learn about DC currents. Changing the limits here is not "wrong" per se, but it may get you an answer to a question you didn't ask!

Sam Bader
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For example, in statistical mechanics, you get different results for systems with spontaneous symmetry breaking, say, for a ferromagnetic, depending on whether you first take the limit $N\rightarrow\infty$ or $H\rightarrow 0$ when calculating average magnetization (http://www.encyclopediaofmath.org/index.php/Quasi-averages,_method_of ).

akhmeteli
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